6.2 Structure of colored, static black holes

We will briefly summarize research in three areas in which the isolated horizon framework
has been used to illuminate the structure of static, colored black holes and associated
solitons7.

6.2.1 Horizon mass

Let us begin with Einstein–Yang–Mills theory considered in the last Section 6.1. As we saw,
the ADM mass fails to be a good measure of the horizon mass for colored black holes. The
failure of black hole uniqueness theorems also prevents the isolated horizon framework from
providing a canonical notion of horizon mass on the full phase space. However, one can repeat the
strategy used for dilatonic black holes to define horizon mass unambiguously for the static
solutions [76, 25].

Consider a connected component of the known static solutions, labelled by . Using for
the surface gravity of the properly normalized static Killing vector, in this sector one can construct a
live vector field and obtain a first law. The energy is well-defined on the full phase space and can
be naturally interpreted as the horizon mass for colored black holes with ‘quantum number’ .
The explicit expression is given by

where, following a convention in the literature on hairy black holes [76], we have used

rather than the surface gravity of static solutions. Now, as in the Einstein–Maxwell case, the
Hamiltonian generating evolution along contains only surface terms: and is
constant on each connected, static sector if coincides with the static Killing field on that sector. By
construction, our has this property for the -sector under consideration. Now, in the
Einstein–Maxwell case, since there is no constant with the dimension of energy, it follows that
the restriction of to the static sector must vanish. The situation is quite different in
Einstein–Yang–Mills theory where the Yang–Mills coupling constant provides a scale. In units,
. Therefore, we can only conclude that

for some -dependent constant . As the horizon radius shrinks to zero, the static
solution [129, 182] under consideration tends to the solitonic solution with the same ‘quantum numbers’
. Hence, by taking this limit, we conclude . Therefore, on any -static
sector, we have the following interesting relation between the black hole and solitonic solutions:

Thus, although the main motivation behind the isolated horizon framework was to go beyond globally
time-independent situations, it has led to an interesting new relation between the ADM masses of black
holes and their solitonic analogs already in the static sector.

The relation (80) was first proposed for spherical horizons in [76], verified in [74], and extended to
distorted horizons in [25]. It provided impetus for new work by mathematical physicists working on
colored black holes. The relation has been confirmed in three more general and non-trivial
cases:

6.2.2 Phenomenological model of colored black holes

Isolated horizon considerations suggested the following simple heuristic model of colored black holes [20]: Acolored black hole with quantum numbersshould be thought of as ‘bound states’ of a ordinary (colorless)black hole and a soliton with color quantum numbers, where can be more general than considered
so far. Thus the idea is that an uncolored black hole is ‘bare’ and becomes ‘colored’ when ‘dressed’ by the
soliton.

The mass formula (80) now suggests that the total ADM mass has three components: the mass
of the bare horizon, the mass of the colored soliton, and a binding energy given by

If this picture is correct, being gravitational binding energy, would have to be negative. This
expectation is borne out in explicit examples. The model has several predictions on the qualitative behavior
of the horizon mass (77), the surface gravity, and the relation between properties of black holes and
solitons. We will illustrate these with just three examples (for a complete list with technical caveats,
see [20]):

For any fixed value of and of all quantum numbers except , the horizon mass and
surface gravity decrease monotonically with .

For fixed values of all quantum numbers , the horizon mass is non-negative,
vanishing if and only if vanishes, and increases monotonically with . is positive
and bounded above by .

For fixed , the binding energy decreases as the horizon area increases.

The predictions for fixed have recently been verified beyond spherical symmetry: for the distorted,
axially symmetric Einstein–Yang–Mills solutions in [133] and for the distorted ‘dipole’ solutions in
Einstein–Yang–Mills–Higgs solutions in [109]. Taken together, the predictions of this model can account for
all the qualitative features of the plots of the horizon mass and surface gravity as functions of the horizon
radius and quantum numbers. More importantly, they have interesting implications on the stability
properties of colored black holes.

One begins with an observation about solitons and deduces properties of black holes. Einstein–Yang–Mills
solitons are known to be unstable [174]; under small perturbations, the energy stored in the ‘bound state’
represented by the soliton is radiated away to future null infinity . The phenomenological model
suggests that colored black holes should also be unstable and they should decay into ordinary black holes,
the excess energy being radiated away to infinity. In general, however, even if one component of a bound
system is unstable, the total system may still be stable if the binding energy is sufficiently large.
An example is provided by the deuteron. However, an examination of energetics reveals that
this is not the case for colored black holes, so instability should prevail. Furthermore, one can
make a few predictions about the nature of instability. We summarize these for the simplest
case of spherically symmetric, static black holes for which there is a single quantum number
:

All the colored black holes on the th branch have the same number (namely, ) of
unstable modes as the th soliton. (The detailed features of these unstable modes can differ
especially because they are subject to different boundary conditions in the two cases.)

For a given , colored black holes with larger horizon area are less unstable. For a given
horizon area, colored black holes with higher value of are more unstable.

The ‘available energy’ for the process is given by

Part of it is absorbed by the black hole so that its horizon area increases and the rest is radiated away
to infinity. Note that can be computed knowing just the initial configuration.

In the process the horizon area necessarily increases. Therefore, the energy radiated to infinity is
strictly less than .

Expectation 1 of the model is known to be correct [173]. Prediction 2 has been shown to be correct in
the , colored black holes in the sense that the frequency of all unstable modes is a decreasing
function of the area, whence the characteristic decay time grows with area [182, 49]. To our
knowledge a detailed analysis of instability, needed to test Predictions 3 and 4 are yet to be
made.

Finally, the notion of horizon mass and the associated stability analysis has also provided an
‘explanation’ of the following fact which, at first sight, seems puzzling. Consider the ‘embedded Abelian
black holes’ which are solutions to Einstein–Yang–Mills equations with a specific magnetic charge .
They are isometric to a family of magnetically charged Reissner–Nordström solutions and the isometry
maps the Maxwell field strength to the Yang–Mills field strength. The only difference is in the form of the
connection; while the Yang–Mills potential is supported on a trivial bundle, the Maxwell potential
requires a non-trivial bundle. Therefore, it comes as an initial surprise that the solution is stable in
the Einstein–Maxwell theory but unstable in the Einstein–Yang–Mills theory [50, 59]. It turns
out that this difference is naturally explained by the WIH framework. Since the solutions are
isometric, their ADM mass is the same. However, since the horizon mass arises from Hamiltonian
considerations, it is theory dependent: It is lower in the Einstein–Yang–Mills theory than in the
Einstein–Maxwell theory! Thus, from the Einstein–Yang–Mills perspective, part of the ADM mass is carried
by the soliton and there is positive which can be radiated away to infinity. In the
Einstein–Maxwell theory, is zero. The stability analysis sketched above therefore
implies that the solution should be unstable in the Einstein–Yang–Mills theory but stable in the
Einstein–Maxwell theory. This is another striking example of the usefulness of the notion of the horizon
mass.

6.2.3 More general theories

We will now briefly summarize the most interesting result obtained from this framework in
more general theories. When one allows Higgs or Proca fields in addition to Yang–Mills, or
considers Einstein–Skyrme theories, one acquires additional dimensionful constants which trigger
new phenomena [48, 179, 182]. One of the most interesting is the ‘crossing phenomena’ of
Figure 11 where curves in the ‘phase diagram’ (i.e., a plot of the ADM mass versus horizon radius)
corresponding to the two distinct static families cross. This typically occurs in theories in which there is
a length scale even in absence of gravity, i.e., even when Newton’s constant is set equal to
zero [154, 20].

Figure 11:

The ADM mass as a function of the horizon radiusin theories with a built-innon-gravitational length scale. The schematic plot shows crossing of families labelled byandat.

In this case, the notion of the horizon mass acquires further subtleties. If, as in the Einstein–Yang–Mills
theory considered earlier, families of static solutions carrying distinct quantum numbers do not cross, there
is a well-defined notion of horizon mass for each static solution, although, as the example of
‘embedded Abelian solutions’ shows, in general its value is theory dependent. When families
cross, one can repeat the previous strategy and use Equation (77) to define a mass
along each branch. However, at the intersection point of the th and th
branches, the mass is discontinuous. This discontinuity has an interesting implication. Consider the
closed curve in the phase diagram, starting at the intersection point and moving along the
th branch in the direction of decreasing area until the area becomes zero, then moving along
to the th branch and moving up to the intersection point along the th branch (see
Figure 11). Discontinuity in the horizon mass implies that the integral of along this
closed curve is non-zero. Furthermore, the relation between the horizon and the soliton mass
along each branch implies that the value of this integral has a direct physical interpretation:

This is a striking prediction because it relates differences in masses of solitons to the knowledge of horizon
properties of the corresponding black holes! Because of certain continuity properties which
hold as one approaches the static Einstein–Yang–Mills sector in the space to static solutions to
Einstein–Yang–Mills–Higgs equations [75], one can also obtain a new formula for the ADM masses of
Einstein–Yang–Mills solitons. If for the black hole solutions of this theory is integrable over
the entire positive half line, one has [20]:

Both these predictions of the phenomenological model [20] have been verified numerically in the spherically
symmetric case [75], but the axi-symmetric case is still open.